Abstract

We investigate the use of a Genetic Algorithm (GA) to design a set of photonic crystals (PCs) in one and two dimensions. Our flexible design methodology allows us to optimize PC structures for specific objectives. In this paper, we report the results of several such GA-based PC optimizations. We show that the GA performs well even in very complex design spaces, and therefore has great potential as a robust design tool in a range of PC applications.

Top: Electric field amplitude of the resonant mode in a slab-waveguide uniform quarter-wave-stack cavity, with a half-wavelength thick central spacer, with Q = 85. Bottom: Electric field amplitude of the resonant mode of our GA-optimized cavity, with Q = 6510. The modal shape at the slab center closely resembles the modal profile of our 1D simulation in Fig 2. In FDTD program units, the widths of the high-index layers (starting from the central spacer) were rhigh = {16, 7, 10, 9, 9, 16, 6, 5, 13, 7} and the widths of the air gaps were rlow = {9, 10, 10, 10, 9, 7, 6, 6, 7}. Both: The thickness of both structures in the vertical direction were fixed at 40 units, and refractive indices are nhigh = 3.30 and nlow = 1.

Left: Tiled unit cells in a hexagonal lattice. Right: Enlarged unit cell, showing the hexagonal arrangement of the nine air holes within each unit cell. The radii of the nine holes are used to encode the chromosomes in our optimization. The white holes with the dotted outlines are not part of the displayed unit cell, but belong to the adjacent cells. Top: Brillouin zone (white hexagon), irreducible Brillouin zone (blue triangle) and Γ, K, and M reciprocal lattice points. The Brillouin zone has the same shape as that of a regular triangular lattice (i.e. a hexagon). However, since the lattice constants in the real-space lattice are longer by a factor of 3 than those of the underlying triangular lattice (as a result of the supercell), the reciprocal-space vectors are correspondingly shorter by a factor of 3.

Fitness (gap-to-midgap ratio at K-point of the band diagram) of maximally-fit structure of each generation for 100 generations. Each line in this figure represents one simulation run of our algorithm. Different runs of the algorithm take different optimization paths, but eventually converge to an optimal solution within approximately 80 generations. The maximum fitness is a monotonically non-decreasing function due to cloning (see section 3). A general increase in fitness arises as a result of various genetic operations (selection, mating, mutation).

Table (a) shows the optimal PC structures predicted by 4 runs of our Genetic Algorithm. The unit cell for each structure is depicted by the yellow bounding box with dimensions 3a × 3 √3a/2, and the unit cell hole radii are listen in the third column. A sample band diagram (for Run 4) is shown in (c). The optimized K-point TE gap, calculated as the ratio of the size of the gap to the midgap value, was found to be ≃ 72%. More bands are shown in (c) to account for the folding of bands due to the supercell. The corresponding K-point gap for a triangular lattice with uniform air holes (r/a = 0.4445) is shown in (d) for reference. The band diagram for the uniform triangular lattice (d) was calculated without a supercell approximation, so the unit cell is three times smaller than for (c). This also implies that the normalized frequencies at the band gap are three times larger, and k-segments on the horizontal axis are three times bigger than for (c).

Table (a) shows the PC structures with a maximized M-point gap, predicted by 4 runs our Genetic Algorithm for unit cells (yellow boxes) with size 3a × 3 √3a/2 for all four runs. (c) shows a sample band diagram (for Run 4). The GA-designed structures have a maximized M-point gap of 64%, which is higher than the M-point gap of 55% of a reference uniform triangular lattice (d). The uniform triangular lattice band diagram (d) was calculated without a supercell approximation, so k-space segments shown on the horizontal axis are 3 times larger than for (c), and normalized frequencies are also 3 times larger than for (c).

Genetic Algorithm prediction of PC structures that have optimally matched E and H fields, for the lowest 4 bands, at the K point. The E-fields are shown for structures with dielectric rods, that have a TM bandgap, while the H-fields are shown for structures with air holes, that have a TE bandgap. The displayed fields are in the direction aligned with the rods. The fields for the lowest 3 bands are very well matched, but begin to deviate significantly from each other at band 4.